It definitely does have the meaning of making an analog signal discrete, and DVDs are digital in that sense.
There's also the sense of serializing or enumerating something into a format more flexible or amenable to computer manipulation. More briefly: to put it on a computer.

For some things it's super obvious: a vinyl record has a groove you can see, and you digitize it by sampling the output of the wiggle: analog.
The contents of an SD card are invisible and the only way to observe them is via a computer, where they're already presented in a flexible format: not analog.

Then you have fuzzy things: a printed picture that was taken with a digital camera. Grab a magnifying glass and you can see that it's got blocks of quantized color, and isn't continuous. You digitize the photo by shining light on it and capturing the bounce.
Optical media is a disc with a groove in it that you read by shining a light on it to capture the bounce. Instead of a horizontal wiggle though it's pits creating an interference pattern with a laser.
It's digital because it's quantized information that it stores. It needs to be digitized to work with easily because the format is often read-only or write once. It's analog because the pits themselves are continuous and the system spends surprising amount of nuance correcting errors from things like fingerprints and dust.

Books are plainly analog, even though they're composed almost entirely of discrete, quantized units of structured information. Sometimes even with a lookup table and index for faster search!

Magnetic tape stores analog audio by varying the strength of a magnet with the audio signal. It stores digital information by only setting the magnet to specific strengths. Same for VHS or cable Internet.

To make a long ramble short: digitize also means to stick it on a computer, owing to efforts to digitize things often being focused on items that aren't continuous, like text.